Integrand size = 21, antiderivative size = 137 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=-\frac {a d \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) (d \sec (e+f x))^{-1+n} \sin (e+f x)}{f (1-n) \sqrt {\sin ^2(e+f x)}}+\frac {b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right ) (d \sec (e+f x))^n \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}} \]
-a*d*hypergeom([1/2, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*(d*sec(f*x+e))^( -1+n)*sin(f*x+e)/f/(1-n)/(sin(f*x+e)^2)^(1/2)+b*hypergeom([1/2, -1/2*n],[1 -1/2*n],cos(f*x+e)^2)*(d*sec(f*x+e))^n*sin(f*x+e)/f/n/(sin(f*x+e)^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\frac {\csc (e+f x) \left (a (1+n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\sec ^2(e+f x)\right )+b n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sec ^2(e+f x)\right )\right ) (d \sec (e+f x))^n \sqrt {-\tan ^2(e+f x)}}{f n (1+n)} \]
(Csc[e + f*x]*(a*(1 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/ 2, Sec[e + f*x]^2] + b*n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[ e + f*x]^2])*(d*Sec[e + f*x])^n*Sqrt[-Tan[e + f*x]^2])/(f*n*(1 + n))
Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4274, 3042, 4259, 3042, 3122}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a+b \sec (e+f x)) (d \sec (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^ndx\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle a \int (d \sec (e+f x))^ndx+\frac {b \int (d \sec (e+f x))^{n+1}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^ndx+\frac {b \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{n+1}dx}{d}\) |
\(\Big \downarrow \) 4259 |
\(\displaystyle a \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n \int \left (\frac {\cos (e+f x)}{d}\right )^{-n}dx+\frac {b \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n \int \left (\frac {\cos (e+f x)}{d}\right )^{-n-1}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{d}\right )^{-n}dx+\frac {b \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{d}\right )^{-n-1}dx}{d}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {b \sin (e+f x) (d \sec (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right )}{f n \sqrt {\sin ^2(e+f x)}}-\frac {a d \sin (e+f x) (d \sec (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n) \sqrt {\sin ^2(e+f x)}}\) |
-((a*d*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec [e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (b*H ypergeometric2F1[1/2, -1/2*n, (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^ n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])
3.8.79.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^(n - 1)*((Sin[c + d*x]/b)^(n - 1) Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
\[\int \left (d \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )d x\]
\[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )\, dx \]
\[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
Timed out. \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx=\int \left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]